Measurement of atomic force microscope cantilever
spring constants (k) is essential for many of
the applications of this versatile instrument.Numerous
techniques to measure k have been proposed.Of these, the thermal noise and Sader methods,
stand out as being widely applicable and relatively user-friendly, providing an in situ, non-destructive, fast measurement of k for a cantilever independent of its material or
coating.Such advantages recommend these
methods for widespread use.An impediment
thereto are the significant complications involved in the initial implementation of the
methods.Some details of the implementation
are discussed in publications, while others are left unsaid.Here we present a complete, cohesive, and practically-oriented discussion of
implementation of both the thermal noise and Sader methods of measuring cantilever spring
constants.We review relevant theory and
discuss practical experimental means for acquiring the required quantities.We then present results which compare measurements
of k by these two methods over nearly two orders
of magnitude, and we discuss likely origins of both statistical and systematic error for
both methods.In conclusion, we find the two
methods agree to within an average of 4 % over the wide range of cantilevers measured.Given that the methods originate in distinct
physics we find the agreement a compelling argument in favor of the accuracy of both,
suggesting them as practical standards for the field.

The doping dependence of nanoscale electronic
structure in superconducting Bi2Sr2CaCu2O8+d is studied by scanning tunneling microscopy. At all dopings,
the low energy density-of-states modulations are analyzed according to a simple model of
quasiparticle interference and found to be consistent with Fermi-arc superconductivity.
The superconducting coherence peaks, ubiquitous in near-optimal tunneling spectra, are
destroyed with strong underdoping and a new spectral type appears. Exclusively in regions
exhibiting this new spectrum, we find local checkerboard charge
ordering of high energy states, with a wave vector of Q = (+/-2p/4.5a0, 0); (0 ; +/-2p/4.5a0)
+/- 15%. Surprisingly, this spatial ordering of high states coexists with the low energy
Bogoliubov quasiparticle states.

At present, the performance of superconducting
qubits is limited by decoherence. Strong decoherence of phase qubits is associated with
spurious microwave resonators residing within the Josephson junction tunnel barrier. In
this work, we investigate three different fabrication techniques for producing tunnel
junctions that vary the properties of the superconductor-insulator interface. Through
experimental measurements, we characterize the junction and corresponding qubit quality.
We find that there is a strong correlation between the morphology of oxidized base
electrodes and the lowering of subgap currents in the junction I-V characteristics, while
there is no noticeable improvement in the performance of fabricated phase qubits. Thus,
traditional indicators of junction performance may not be enough to determine
qubit performance. However, truly crystalline insulating barriers may be the key to
improving Josephson junction based qubits.

Although Josephson junction qubits show great
promise for quantum computing, the origin of dominant decoherence mechanisms remains
unknown. Improving the operation of a Josephson junction based phase qubit has revealed
microscopic two-level systems or resonators within the tunnel barrier that cause
decoherence. We report spectroscopic data that show a level splitting characteristic of
coupling between a two-state qubit and a two-level system. Furthermore, we show Rabi
oscillations whose coherence amplitude is significantly degraded
by the presence of these spurious microwave resonators. The discovery of these resonators
impacts the future of Josephson qubits as well as existing Josephson technologies.

Increasing demands on nanometer-scale properties
of oxide tunnel barriers necessitate a consistent means to assess them on these length
scales. Conducting atomic force microscopy (CAFM) is a promising technique both for
understanding connections between nanoscale tunnel barrier characteristics and macroscopic
device performance as well as for rapid qualitative evaluation of new fabrication methods
and materials. Here we report CAFM characterization of aluminum oxide (AlOx) barriers to
be used in Josephson-junction qubits, with a particular emphasis on developing
reproducible imaging conditions and appropriate interpretation. We find that control of
the imaging force is a critical factor for reproducibility. We imaged the same sample on
the same day with the same cantilever varying only the imaging force between scans.
Statistical properties compiled from the resulting current maps varied approximately
exponentially with imaging force, with typical currents increasing by two orders of
magnitude for only a factor of 5 increase in imaging force. Given appropriate control of
the imaging force, scan to scan variation of the current recorded at the same location was
approximately ±0.5 Iavg, which establishes a criterion for statistical
reproducibility of CAFM measurements. We further find that the appropriate interpretation
for CAFM (under most imaging conditions), is as a probe of local propensity for insulator
breakdown. Samples stored in air for weeks before study showed current features with
oxidation times of order minutes. This indicates that these features were created by the
scanning of the tip, and thus represent local pinhole susceptible regions. We finally
present results for several AlOx samples showing that under appropriate imaging conditions
significant sample to sample variation is observed, thus demonstrating the potential of
this technique to qualitatively assess and facilitate understanding of potential qubit
tunnel barrier devices.

Current-biased Josephson junctions are prime
candidates for the realization of quantum bits; however, a present limitation is their
coherence time. In this paper it is shown qualitatively that quasiparticles create
decoherence. We can decrease the number of quasiparticles present in the junctions by two
methods - reducing the creation rate with current shunts and increasing the depletion rate
with normal-metal traps. Experimental data demonstrate that both methods are required to
significantly reduce the number of quasiparticles and increase the system's coherence. We
conclude that these methods are effective and that the design of Josephson-junction qubits
must consider the role of quasiparticles.

Low temperature scanning tunneling microscopy
(STM) of various samples of the high temperature superconductor Bi2Sr2CaCu2O8+d consistently reveals the presence of quasi-particle
scattering resonances, similar both spectro-scopically and spatially to those observed
around Zn atoms in Zn-doped BSCCO. As the resonances appear at energies indicative of
nearly unitary scattering (~0.5 meV) and are always accompanied by topographic depression
of the surface Bi atom around which they are centered, we postulate that the source of
scattering may be Cu vacancies in the CuO2 plane, Such resonances should thus
provide a simpler test case for theoretical models than those created by Zn or Ni
substitution.

Scanning tunneling spectroscopy of Bi2Sr2CaCu2O8+d reveals weak, incommensurate, spatial modulations in the
tunneling conductance. When images of these energy-dependent modulations are Fourier
analyzed the dispersion of their wave vectors can be determined. Comparison of the
dispersions with angle-resolved photoemission indicates that quasiparticle interference,
due to elastic scattering between specific regions of the Fermi surface, provides a
consistent explanation for the conductance modulations.

We calculate for the current-biased Josephson
junction the decoherence of the qubit state from noise and dissipation. The effect of
dissipation can be entirely accounted for through a semiclassical noise model that
appropriately includes the effect of zero-point and thermal fluctuations from dissipation.
The magnitude and frequency dependence of this dissipation can be fully evaluated with
this model to obtain design constraints for small decoherence. We also calculate
decoherence from spin echo and Rabi control sequences and show they are much less
sensitive to low-frequency noise than for a Ramsey sequence. We predict small decoherence
rates from 1/f noise of charge, critical current, and flux based on noise measurements in
prior experiments. Our results indicate this system is a good candidate for a solid-state
quantum computer.

Scanning tunneling spectroscopy of the high-Tc
superconductor Bi2Sr2CaCu2O8+d
reveals weak, incommensurate, spatial modulations in the tunneling conductance. Images of
these energy-dependent modulations are Fourier analyzed to yield the dispersion of their
wavevectors. Comparison of the dispersions with photoemission spectroscopy data indicates
that quasiparticle interference, due to elastic scattering between characteristic regions
of momentum-space, provides a consistent explanation for the conductance modulations,
without appeal to another order parameter. These results refocus attention on
quasiparticle scattering processes as potential explanations for other incommensurate
phenomena in the cuprates. The momentum-resolved tunneling spectroscopy demonstrated here
also provides a new technique with which to study quasiparticles in correlated materials.

Granular superconductivity occurs when
microscopic superconducting grains are separated by non-superconducting regions; Josephson
tunnelling between the grains establishes the macroscopic superconducting state. Although
crystals of the copper oxide high-transition-temperature (high-Tc) superconductors are not
granular in a structural sense, theory suggests that at low levels of hole doping the
holes can become concentrated at certain locations resulting in hole-rich superconducting
domains. Granular superconductivity arising from tunnelling between such domains would
represent a new view of the underdoped copper oxide superconductors. Here we report
scanning tunnelling microscope studies of underdoped Bi2Sr2CaCu2O8+d that reveal an apparent segregation of the electronic
structure into superconducting domains that are ~3 nm in size (and local energy gap <50
meV), located in an electronically distinct background. We used scattering resonances at
Ni impurity atoms as 'markers' for local superconductivity; no Ni resonances were detected
in any region where the local energy gap Delta >50 +or- 2.5 meV. These observations
suggest that underdoped Bi2Sr2CaCu2O8+d is a mixture of two different short-range electronic orders
with the long-range characteristics of a granular superconductor.

Scanning tunneling microscopy is used to image
the additional quasi-particle states generated by quantized vortices in the high critical
temperature superconductor Bi2Sr2CaCu2O8+d. They exhibit a copper-oxygen bond-oriented
"checkerboard" pattern, with four unit cell (4a0) periodicity and a
~30 angstrom decay length. These electronic modulations may be related to the magnetic
field-induced, 8a0 periodic, spin density modulations with decay length of ~70
angstroms recently discovered in La1.84Sr0.16CuO4. The
proposed explanation is a spin density wave localized surrounding each vortex core.
General theoretical principles predict that, in the cuprates, a localized spin modulation
of wavelength lambda should be associated with a corresponding electronic modulation of
wavelength lambda /2, in good agreement with our observations.

Wepresent scanning tunneling spectroscopy
measurements of the CuO chain plane in YBa2Cu3O6 + x,
showing a ~25 meV gap in the local density of states (LDOS) filled by numerous
intragap resonances: intense peaks in LDOS spectra associated with one-dimensional,
Friedel-like oscillations. We discuss how these phenomena shed light on recent results
from other probes, as well as their implications for phenomena in the superconducting CuO2
plane.

The parent compounds of the copper oxide
high-transition-temperature (high-Tc) superconductors are unusual
insulators (so-called Mott insulators). Superconductivity arises when they are 'doped'
away from stoichiometry. For the compound Bi2Sr2CaCu2O8+x,
doping is achieved by adding extra oxygen atoms, which introduce positive charge carriers
('holes') into the CuO2 planes where the superconductivity is believed to
originate. Aside from providing the charge carriers, the role of the oxygen dopants is not
well understood, nor is it clear how the charge carriers are distributed on the planes.
Many models of high-Tc superconductivity accordingly assume that the
introduced carriers are distributed uniformly, leading to an electronically homogeneous
system as in ordinary metals. Here we report the presence of an electronic inhomogeneity
in Bi2Sr2CaCu2O8+x, on the basis of
observations using scanning tunnelling microscopy and spectroscopy. The inhomogeneity is
manifested as spatial variations in both the local density of states spectrum and the
superconducting energy gap. These variations are correlated spatially and vary on the
surprisingly short length scale of 14 Å. Our analysis suggests that this
inhomogeneity is a consequence of proximity to a Mott insulator resulting in poor
screening of the charge potentials associated with the oxygen ions left in the BiO plane
after doping, and is indicative of the local nature of the superconducting state.

Magnetic interactions and magnetic impurities are
destructive to superconductivity in conventional superconductors. By contrast, in some
unconventional macroscopic quantum systems (such as superfluid 3He and
superconducting UGe2), the superconductivity (or superfluidity) is actually
mediated by magnetic interactions. A magnetic mechanism has also been proposed for
high-temperature superconductivity. Within this context, the fact that magnetic Ni
impurity atoms have a weaker effect on superconductivity than non-magnetic Zn atoms in the
high-Tc superconductors has been put forward as evidence supporting a
magnetic mechanism. Here we use scanning tunnelling microscopy to determine directly the
influence of individual Ni atoms on the local electronic structure of Bi2Sr2CaCu2O8+d. At each Ni site we observe two d-wave impurity states of
apparently opposite spin polarization, whose existence indicates that Ni retains a
magnetic moment in the superconducting state. However, analysis of the impurity-state
energies shows that quasiparticle scattering at Ni is predominantly non-magnetic.
Furthermore, we show that the superconducting energy gap and correlations are unimpaired
at Ni. This is in strong contrast to the effects of non-magnetic Zn impurities,
which locally destroy superconductivity. These results are consistent with predictions for
impurity atom phenomena derived from a magnetic mechanism.

We have previously reported a 1.36-nm DOS
modulation in the CuO chains on the surface of cold- cleaved, atomically-resolved YBCO. We
have recently completed new experiments in which we obtained detailed spectroscopic data
on the same crystal plane, thereby confirming and extending our earlier work. Here we
present data showing twin boundaries and steps, and focusing on the effects of these
structures on the CuO chain DOS modulations.

Although the crystal structures of the copper
oxide high-temperature superconductors are complex and diverse, they all contain some
crystal planes consisting of only copper and oxygen atoms in a square lattice:
superconductivity is believed to originate from strongly interacting electrons in these
CuO2 planes. Substituting a single impurity atom for a copper atom strongly
perturbs the surrounding electronic environment and can therefore be used to probe
high-temperature superconductivity at the atomic scale. This has provided the motivation
for several experimental and theoretical studies. Scanning tunnelling microscopy (STM) is
an ideal technique for the study of such effects at the atomic scale, as it has been used
very successfully to probe individual impurity atoms in several other systems. Here we use
STM to investigate the effects of individual zinc impurity atoms in the high-temperature
superconductor Bi2Sr2CaCu2O8+d.
We find intense quasiparticle scattering resonances at the Zn sites, coincident with
strong suppression of superconductivity within 15 Å of the scattering sites. Imaging of
the spatial dependence of the quasiparticle density of states in the vicinity of the
impurity atoms reveals the long-sought four-fold symmetric quasiparticle 'cloud' aligned
with the nodes of the d-wave superconducting gap which is believed to characterize
superconductivity in these materials.

High-resolution, low-temperature scanning
tunneling microscopy and spectroscopy on Bi2Sr2CaCu2O8+d reveal the existence of large numbers of scattering centers
in this material. The spatial and spectroscopic characteristics of these features are
consistent with theories of quasiparticle scattering from atomic scale impurities in a
d-wave superconductor. These characteristics include breaking of local particle-hole
symmetry and an inverse square dependence of their local density-of-states (LDOS) on
distance from the scattering center. Furthermore, these observations identify a source for
the anomalously high levels of low-energy excitations in Bi2Sr2CaCu2O8+d at low-temperatures.

We report on the results of a search for
superconductivity in Li. We find no evidence for the predicted transition to
superconductivity at any temperature down to 5 mK in magnetic fields down to 0.4 microT.
However, an unexpected Curie- Weiss temperature dependence in the magnetic susceptibility
is observed. We discuss the possibility that this signal arises from the Li itself, the
possibility that it arises from Kondo behavior, and the implications of the effect for the
predicted Tc of Li.